Mahler’s measures on function spaces

Abstract

Recently, I. Pritsker considered a Bergman-space version of Mahler’s measure, and obtained many nice properties such as the arithmetic nature, relation with asymptotic zero distribution, etc. (Illinois J. Math. 52 (2009) 347–363). In this paper, we define a Fock-space analogue of Mahler’s measure, and show a similar version of Lehmer’s conjecture. Inspired by this result, we establish an equivalent form of Lehmer’s conjecture. Also, this consideration is done on weighted Bergman spaces. However, it is shown that in this case the corresponding form of Lehmer’s conjecture fails. In addition, we give an affirmative answer to an approximation question raised by I. Pritsker (Illinois J. Math. 52 (2009) 347–363).